Verify that \frac{d}{d z}|z|=\theta(z)-\theta(-z) where 0(z) = 1, if z > 0, and 0, if z < 0. Further, verify that \frac{d^{2}}{d z^{2}}|z|=2 \delta(z) Also, argue that, for a well

defined function f(2), the replacement f(z) \delta(z)=f(0) \delta(z) is justified. Using Eq. (1), Eq. (2), and Eq. (3), verify (by substituting the solution intothe differential equation) that g(z)=\frac{1}{2 k} e^{-k|z|} is a particular solution of the differential equation -\left(\frac{d^{2}}{d z^{2}}-k^{2}\right) g(z)=\delta(z)